tariffs lead to inflation; is this the same as inflation from supply constraints or demand drive, given that the tariff is removed from the money supply? How should the fed be thinking about this in the context of raising interest rates if they use inflation a gauge of how much to hike?
Across the board tariffs (ex: sales tax) can result in higher prices or "inflated" prices. This is usually not what economists refer to when they say inflation. Inflation that the fed is concerned with, is an across the board increase in all goods as a result from changes in the money supply. Something, that the fed can directly influence, whereas the fed has no control over taxes.
Fiscal policy (Government) and monetary policy (The Fed) can sometimes coordinate actions for a desired result.
For instance: Following the 2007/8 financial crisis, the united states paired subsidies and incentives ,effectively lowering sales tax (fiscal policy), with low interest rates (monetary policy) to stimulate the economy.
You can imagine that a coordinated policy concerned with high inflation rates might do the opposite.